# Does an electron have kinetic energy when attached to a proton?

• B
• Lolicon
That is, if we could measure the kinetic energy of all the electrons in the box at every moment, we would get a number that is not zero but is smaller than the total energy of the system.The expected value of kinetic energy is smaller than the total energy of the system.f

#### Lolicon

does an electron have kinetic energy when attached to a proton? if not, what is it transformed into?

does an electron have kinetic energy when attached to a proton? if not, what is it transformed into?
A proton and electron bound together is called a hydrogen atom. The hydrogen atom has a set of possible energies, each of which is the sum of the kinetic energy and potential energy.

• vanhees71 and Dale
Yes, it has KE

There is a nice result called the Virial theorem which says that if the potential energy of interaction between two bound particles goes as ##V(r) = \lambda r^n##, then the potential and kinetic energies of that system are related via ##2 \langle T \rangle = n \langle V \rangle ##. For an electrostatic interaction that potential has ##n=-1##, so ##2\langle T \rangle = - \langle V \rangle##. Equivalently, the total energy is ##E = \langle T \rangle + \langle V \rangle = - \langle T \rangle = \langle V \rangle/2##.

The energy eigenvalues of a hydrogen atom look like$$E = \frac{-13.6 \text{eV}}{n^2}$$For instance at the ground state, ##n=1##, then ##E = -13.6 \text{eV}##, ##\langle V \rangle = -27.2 \text{eV}## and ##\langle T \rangle = 13.6 \text{eV}##. As you go up energy levels, the potential energy and total energy increase, whilst the kinetic energy decreases.

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• • Lolicon, JD_PM, berkeman and 1 other person
The energy eigenvalues of a hydrogen atom look like$$E = \frac{-13.6 \text{eV}}{n^2}$$For instance at the ground state, ##n=1##, then ##E = -13.6 \text{eV}##, ##V = -27.2 \text{eV}## and ##T = 13.6 \text{eV}##. As you go up energy levels, the potential energy and total energy increase, whilst the kinetic energy decreases.

Those are, of course, the expected values of ##T## and ##V## for a QM system.

• JD_PM and etotheipi
Those are, of course, the expected values of ##T## and ##V## for a QM system.

Thanks, give me two seconds and I'll put the langles and the rangles where they belong • JD_PM and vanhees71
Possible nitpick about terminology: "expectation value" is the term that I've always seen in English-language QM textbooks, journal articles, etc., at least in the US.

To me, "expected value" has the connotation "It has a definite value, and this is what I expect it to be." Of course, that's not what we're talking about here.

• weirdoguy and etotheipi
Whenever an electron is moving in some limited space, as when orbiting an atom or put in a "box", it has a nonzero expectation value of kinetic energy.